1. ©2002 Regents of University of Minnesota Fluidization of 1204 Spheres 1 Direct Numerical Simulation (DLM) of 1204 Spheres in a Slit Bed zSphere Diameter 0.25” yCompare computed bed expansion with the observed 0.275 in. 8 in. 58.5 in. Pan, Sarin, Joseph, Glowinski & Bai 1999

2. ©2002 Regents of University of Minnesota Fluidization of 1204 Spheres 2 Crystal configuration 1204 Particles 25 20 15 10 5 0 02468 SimulationExperiment

3. ©2002 Regents of University of Minnesota Fluidization of 1204 Spheres 3 V = 2.0 : Particle position at t = 5.625. The maximal particle Reynolds number is 1383. The maximal averaged particle Reynolds number is 130. 1204 Particles Experiment 02468 25 20 15 10 5 0 Simulation

4. ©2002 Regents of University of Minnesota Fluidization of 1204 Spheres 4 V = 3.0 : Particle position at t = 20 The maximal particle Reynolds number is 1142. The maximal averaged particle Reynolds number is 131. 1204 Particles Experiment 25 20 15 10 5 0 02468 Simulation

5. ©2002 Regents of University of Minnesota Fluidization of 1204 Spheres 5 V = 3.5 : Particle position at t = 17.238 The maximal particle Reynolds number is 1671. The maximal averaged particle Reynolds number is 236. 1204 Particles Experiment 02468 25 20 15 10 5 0 Simulation

6. ©2002 Regents of University of Minnesota Fluidization of 1204 Spheres 6 V = 4.0 : Particle position at t = 32 The maximal particle Reynolds number is 1965. The maximal averaged particle Reynolds number is 276. 1204 Particles Simulation Experiment 02468 25 20 15 10 5 0

7. ©2002 Regents of University of Minnesota Fluidization of 1204 Spheres 7 V = 4.5 : Particle position at t = 31 The maximal particle Reynolds number is 1859. The maximal averaged particle Reynolds number is 292. 1204 Particles Experiment 02468 25 20 15 10 5 0 Simulation

8. ©2002 Regents of University of Minnesota Fluidization of 1204 Spheres 8 Bed Expansion Richardson-Zaki V(  ) A d = 1/4” Superficial Inlet Velocity

9. ©2002 Regents of University of Minnesota Fluidization of 1204 Spheres 9 Blow Out Velocity

10. ©2002 Regents of University of Minnesota Fluidization of 1204 Spheres 10 Bed Height vs. Fluidizing Velocity for Both Experiment and Simulation zFor the monodispersed case studied in simulation (d = 0.635cm) H s = 4.564/(1-  ) zThe mean sphere size for the polydisperse case studied in the experiments is slightly larger (d = 0.6398cm) and H e = 4.636/(1-  )

11. ©2002 Regents of University of Minnesota Fluidization of 1204 Spheres 11 Data from Previous Slide Plotted in a log-log Plot and D is the tube radius. In our experiments and simulations Re is confined to the range for which n = 2.39.  The slopes of the straight line are given by the Richardson- Zaki n = 2.39. The blow-out velocities V s (0) and V e (0) are defined as the intercepts at  = 1. zThe Richardson-Zaki correlation is given by V(  ) = V(0)  n(Re) where V(0) is V when  = 1  V s (  ) = 8.131  2.39 cm/s and  V e (  ) = 10.8  2.39 cm/s.

12. ©2002 Regents of University of Minnesota Fluidization of 1204 Spheres 12 Bed Height vs. Fluidizing Velocity After Shifting by the Ratio of Blow-out Velocities zd 1 = 0.635cm simulation zd 2 = 0.6398cm (average d for experiments) zWalls will increase the drag more in the experiments than in the simulations. Wall correction of Francis [1933] zThe value 1.233 is very close to the shift ratio

13. ©2002 Regents of University of Minnesota Fluidization of 1204 Spheres 13 Slip Velocity is Computed on Data Strings at Nodes U2U2 U1U1 U3U3 The slip velocity for U 1 - U 2 is zero + noise

14. ©2002 Regents of University of Minnesota Fluidization of 1204 Spheres 14 Transitions Between Power Laws Logistic dose curve This curve is fitted to data and is convenient to use with a spread sheet

15. ©2002 Regents of University of Minnesota Fluidization of 1204 Spheres 15 Transitions Between Power Laws Example: Richardson-Zaki Correlation Power law Transition Power law